Kinetic energy proportional to velocity squared in oscillatory motion

Steps:

• Displacement of air particle in sound wave follows sinusoidal pattern: $x = A \sin(\omega t)$.
• Velocity is time derivative: $v = \frac{dx}{dt} = A \omega \cos(\omega t)$.
• Kinetic energy $KE = \frac{1}{2} m v^2 \propto \cos^2(\omega t)$, always positive and oscillating twice per cycle.
• Graph of KE vs. time shows periodic humps without negative values, matching option D.

Why D is correct:

• D depicts $\cos^2(\omega t)$, directly from KE formula, with peaks at zero displacement (max speed).

Why the others are wrong:

• A shows linear increase, ignoring oscillatory nature of wave.
• B mirrors displacement, but KE depends on $v^2$, not position.
• C resembles velocity, which can be negative, unlike always-positive KE.

Final answer: D